1. Field of the Invention
The present invention is directed to a method for computerized design of fuzzy logic rules from training vectors of a training dataset, as well as to an apparatus for implementing the method.
2. Description of the Prior Art
In the framework of so-called "unsharp" logic (fuzzy logic), technical systems are known which make use of so-called if/then rules. The relationships between the input quantities of the rules and the output quantities of the rules are defined by means of membership functions that supply a value between 0 and 1 for every input value and output value. The rules, which are combinations of membership functions, are formulated by persons knowledgeable in this field in some instances. This procedure is referred to as "manual" design because the rules are defined by knowledgeable persons who can describe the technical system in detail in terms of its behavior.
Usually, however, the rules are defined on the basis of training data vectors that contain the output values (specified values) corresponding to various input values (actual values). This procedure is referred to as "automatic" design.
The training data, for example, can be determined by measuring instruments, for example sensors, and describe the real behavior of the system. A technical system can, for example, be a washing machine, a factory procedure, communication equipment, a communication network, etc. The type of system is not critical tot he rule formation procedure.
A large variety of membership functions can be employed in the framework of fuzzy logic. As used herein, a "membership function type" means a type of membership function that is unambiguously described by its form. An overview of various membership function types can be found in Klir et al., Fuzzy Sets and Fuzzy Logic--Theory and Applications, Prentice Hall P T R, New Jersey, ISBN 0-13-101171-5, pp. 97-102, 1995.
Various cluster algorithms are usually employed in the methods for automatic rule design.
What is understood by clustering is a method with which a set X={x.sub.1, . . . , x.sub.n }.OR right..sup.p is grouped in sub-sets (clusters) c.di-elect cons.{2, . . . , n-1} that represent a sub-structure of the set X. The training data vectors of the training dataset that is employed for rule design is referenced x.sub.1, . . . , x.sub.n. The training dataset contains n training data vectors. A partition matrix U.di-elect cons.M=[0, 1].sup.cn describes the division of the set X into the individual clusters. Each element u.sub.ik, i=1, . . . , c; k=1, . . . , n, of the partition matrix U represents the membership of the training data vector x.sub.k .di-elect cons.X to the i.sup.th cluster. The designation u.sub.ik is used below to refer to the membership value of the k.sup.th training data vector x.sub.k to the i.sup.th cluster.
A model known as the fuzzy C-means model (FCM) that is utilized for automatic rule design is known from J. C. Bezdek, Pattern Recognition with Fuzzy Objective Function Algorithms, Plenum Press New York, ISBN 0-306-40671-3, pp. 65-79, 1981.
FCM forms a fuzzy partition M.sub.fcn defined as: ##EQU1##
The FCM is defined as the following problem:
With a given training dataset X, an arbitrary norm .parallel...parallel. on the space .sup.p and a prescribable fuzzy parameter m.di-elect cons.(1, .varies.), the following target function is to be minimized: ##EQU2## A set of centers of membership functions employed is referenced V={v.sub.1, . . . , v.sub.c }.OR right..sup.p.
In the FCM alternating optimization method (FCM-AO) likewise described in Bezdek, cluster centers are initialized with arbitrary vectors v.sub.i .di-elect cons..sup.p, i=1 . . . c. In an iterative method, membership values u.sub.ik for the training data vectors x.sub.k and the cluster centers v.sub.i are determined in alternation dependent on the previously identified membership values u.sub.ik. This iterative method is ended when either a maximum number of iterations has been implemented or when the change of the center of gravity values for preceding iterations is smaller than a prescribable threshold. The membership values u.sub.ik are formed according to the following rule in each iteration for all training data vectors: ##EQU3##
The centers v.sub.i of the membership functions are formed according to the following rule by a determination of center of gravity values for respective membership function: ##EQU4##
Another model known as the possibilistic C-means model (PCM) is known from Krishnapuram et al., A possibilistic approach to clustering, IEEE Transactions on Fuzzy-Systems, Vol. 1, No. 2, pp. 98-110, 1993. The PCM differs from the FCM on the basis of a modified target function. The target function J.sub.FCM of the FCM has a penalty term added to it.
The target function J.sub.PCM in the PCM is formed according to the following rule: ##EQU5##
An auxiliary factor .eta..sub.i is formed according to the following rule: ##EQU6##
The factor K.di-elect cons..sup.+ .backslash.{0} is usually selected as value 1.
The following rule is another possibility for determining the auxiliary factor .eta..sub.i : ##EQU7##
With a prescribable value .alpha..di-elect cons.(0,1), a factor (u.sub.ik).sup..gtoreq..alpha. is determined according to the following rule: ##EQU8##
An alternate method (PCM-AO) for minimizing the target function in PCM is also known from Krishnapuram et al. In the method fundamentally designed the same as in FCM-AO, the membership values u.sub.ik are formed according to the following rule, and it is assumed that .eta..sub.i &gt;0.A-inverted.i is respectively valid for the auxiliary factor: ##EQU9##
The following rule is employed for determining the respective centers v.sub.i of the membership functions in every iteration: ##EQU10##
After the conclusion of this iterative method, centers of the respective membership functions are identified in FCM-AO as well as PCM-AO. The membership functions positioned over the identified centers v.sub.i of the membership functions are interpreted as rules of the fuzzy system.
The elements of the partition matrix U that derives from FCM-AO or from PCM-AO are values of the membership functions EQU u.sub.i :X.fwdarw.[0,1],i=1, . . . , c
for the clusters in X, whereby EQU u.sub.i (x.sub.k)=u.sub.ik, i=1, . . . , c; k=1, . . . , n.
Below, the values {u.sub.i (x.sub.k)} are interpreted as observations of expanded membership functions EQU .mu..sub.i :p.fwdarw.[0,1], .mu..sub.i (x.sub.k)=u.sub.i (x.sub.k)=u.sub.ik, k=1, . . . ,n (10)
The expanded membership functions .eta..sub.i are formed for FCM-AO according to the following rule: ##EQU11##
For PCM-AO, the expanded membership functions .eta..sub.i are determined according to the following rule: ##EQU12##
A considerable disadvantage of the known methods is that the form of the membership functions is fixed both in FCM or in FCM-AO, as well as in PCM or in PCM-AO. A non-convex, bell-shaped function arises in FCM, and the Cauchy function arises in PCM. In FCM-AO as well as in PCM-AO, the method is thus limited to a fixed type of membership function.
Other forms of membership functions and thus other membership function types, can only be obtained by a subsequent approximation of the membership function given this method. This leads to a considerable imprecision of the identified rules.